3 Acknowledgments The first person I would like to thank is my advisor Professor Carlos Florentino for proposing this subject to me and for guiding me throughout this work. His motivation and the time he took with me were very important for me to finish this thesis. I would like to thank Professor Margarida Mendes Lopes for introducing me to Algebraic Geometry and for motivating me to work harder to achieve success. I also wish to thank my parents and my family for providing me with a good atmosphere to work and for caring about me and always being around. Obrigado Pai, Mãe, Joana e Rui! Last but not least, I would like to thank my friends and brothers in arms for being beside me and for their advice and motivation. It is thanks to them that I haven t (completely) despaired! i

7 Abstract This thesis deals with holomorphic fiber bundles over Riemann surfaces and, in particular, over the Riemann sphere P 1. Basing ourselves on Grothendieck s Theorem on the classification of principal bundles over P 1 and after introducing many important results and techniques in the theory of vector bundles, we present and prove a classification of principal bundles with orthogonal or symplectic structure groups over the Riemann sphere. Keywords: Riemann Surface, Riemann Sphere, Vector Bundle, Principal Bundle, Cohomology, Lie Group. v

11 Introduction In short, a Riemann surface is an object that locally behaves as C and where we can define coherently holomorphic functions. One of the most famous examples of such is the Riemann sphere, that may be denoted by P 1 and is homeomorphic to the surface of a sphere, hence the name. A fiber bundle is an object defined over a topological space, in particular a Riemann surface, by connecting a fixed set (fiber) to every point of the latter and satisfying a local trivialization condition. The vast amount of different types of conditions for this local trivialization, provides many different structures that one can study. In particular, we will focus on vector bundles, where the fibers are a vector space, and principal bundles, where we consider the action of a group on the fibers. The main purpose of this work is to study a result on classification of vector bundles and principal bundles over the Riemann sphere, due to Grothendieck. The vector bundle version is a very famous result in Algebraic Geometry and is known simply as Grothendieck s Theorem. However, on the original paper it was a given a more general result for principal bundles with structure group a complex Lie group G, from which the previous result arises as a corollary, when we take GL(n, C) as the structure group. Our goal is to present the more general case in a clear way, going through the appropriate basic concepts that are necessary to understand it. We will also prove it for the particular cases of vector bundles with an holomorphic quadratic form, or a symplectic form defined on its fibers. Through the way, we will study basic results on Riemann surfaces and introduce other concepts such as sheaves, its cohomology groups, and Lie groups. 1

12 2

13 Chapter 1 Vector Bundles Throughout this chapter we will focus on Riemann surfaces and vector bundles. First we will define properly a Riemann surface and give some examples. We will then take a look at the analogous case for greater dimensions. By then, we will be able to give a proper definition of vector bundle, along with the first few examples. Then, we define an important tool to study vector bundles, sheaves, taking the sheaf of sections of a vector bundle as our main example. Furthermore, we define the cohomology groups of a sheaf and we see the long exact sequence they generate from a short exact sequence of sheaves. Then, we present two fundamental results: Riemann-Roch s Theorem and Serre s Duality. The chapter culminates with Grothendieck s Theorem for vector bundles (Theorem 3). 1.1 Riemann Surfaces A Riemann surface is, in a sense, a generalization of the complex plane. It is a 2- dimensional real manifold that is locally homeomorphic to an open set of C and where we can define holomorphic functions coherently. To study a surface locally, we describe it through a well known object, for example the complex plane C, using coordinate charts. Definition 1. A complex coordinate chart, or simply chart, on a topological space S is a homeomorphism φ : U W, where U S and W C are open sets. Remark. It is also usual to use z to refer to the image of a point p S through the map φ. We write z(p) = φ(p) to be more precise, or z = φ(p). Definition 2. A Riemann surface S is a topological space, together with a covering of coordinate charts {φ α : U α W α C, S = α U α }, such that for every pair α, β, φ α φ 1 β 3

14 is an invertible holomorphic function. Remark. From now on, when omitted, S will designate a Riemann surface. Example 1. The projective line P 1, also known as the Riemann sphere, is defined as the quotient (C 2 \ {(0, 0)})/, with (x, y) λ(x, y) = (λx, λy) for any λ C. A point in P 1 is denoted by (x : y), when (x, y) is a representative of the class it defines through. are: To see that this is a Riemann surface, we use two coordinate charts to cover P 1. They U 0 = {(1 : y) : y C} U 1 = {(x : 1) : x C} φ 0 : U 0 C The change of chart is an inversion: φ 1 : U 1 C (1 : y) y (x : 1) x (( φ 0 φ 1 1 (x) = φ 0((x : 1)) = φ 0 1 : 1 )) = 1 x x, x C, which is a holomorphic automorphism on C. Example 2. The torus T 2 = S 1 S 1 is also a Riemann surface. To see this, we consider the lattice generated by 1 and i, (Z + iz) C. Then, we set the torus as the quotient T 2 = R 2 /Z 2 = C/(Z + iz). It is easy to obtain a covering of holomorphic coordinate charts for this case. an example, we suggest charts obtained from the projection into the torus of the sets in {B 1 (0), B 1 ( ), B 1 ( i), B 1 ( i)}, with B 1 (z) = {w C : w z < }. Notice that for each of these sets the projection into the torus is a bijection, as the difference of any two distinct points does not belong to the lattice (Z + iz). Definition 3. A function f : S C is holomorphic if so are its representatives f φ 1 α, for a covering of coordinate charts {φ α : U α C}. Proposition 1. Let S be a connected compact Riemann surface, then the only holomorphic functions on S are the constant functions. Proof. Let f be an holomorphic function over S. So, f is continuous and as S is compact, we conclude that f has a maximum point. maximum point may not be strict. Therefore, f is constant. As By the Maximum Modulus Principle, this If f = 0, then we are done. If not, we can see that f = f 2 f is also holomorphic. Using the Cauchy-Riemann equations for f and f, we conclude that their directional derivatives must be 0. Hence, f is constant and we are done. 4

15 1.2 Complex Manifolds Similarly to a Riemann surface, a n-dimensional complex manifold is an object that locally behaves like C n. Though its behavior is more complex than in the case of Riemann surfaces, the basic definitions will be similar. Definition 4. A n-dimensional complex coordinate chart, or simply chart, on a topological space M is a homeomorphism φ : U W, where U M and W C n are open sets. Definition 5. A n-dimensional complex manifold M is a topological space, together with a covering of coordinate charts {φ α : U α W α C n }, such that for every pair α, β, φ α φ 1 β is an invertible holomorphic function. 1.3 Vector Bundles On this work we intend to study holomorphic vector bundles over Riemann surfaces (in particular, over the Riemann sphere). So, let us define the objects of our study: Definition 6. A holomorphic vector bundle of rank n, E, over a Riemann surface S is a (n + 1)-dimensional complex manifold together with a projection map π : E S, such that: 1. for each point p S, π 1 (p) is a C-vector space of dimension n; 2. for each point p S there is a neighborhood of p, U S, and a homeomorphism ϕ U, such that the following diagram commutes: π 1 (U).. U C n π.. ϕ U.. U p 1 (p 1 is the usual projection on the first coordinate). 3. For every pair of homeomorphisms ϕ U and ϕ V as in 2. and whenever U V, ϕ U ϕ 1 V (U V ) Cn has the form ϕ U ϕ 1 V (U V ) C n : (U V ) Cn (U V ) C n (p, w) (p, A UV (p)w) where A UV : U V GL(n, C) defines a linear isomorphism (these are called transition functions). 5

16 Remark. Vector bundles of rank 1 are called line bundles. On the sequel, unless it is explicit on the context, we will use n to denote the rank of a vector bundle. We will use E x to denote the fiber of a vector bundle E at a point x S. Example 3. (1) The trivial vector bundle S C n over S. (2) The simplest non-trivial line bundles over S are denoted by L p, where p S. To define these line bundles we take a chart φ 0 ( ) = z( ) of a neighborhood U 0 of p, with z(p) = 0, and the open set U 1 = S \ {p}. Furthermore, L p is given by gluing U 0 C and U 1 C through the transition function g 01 = z. Notice that g 01 is invertible, i.e. never vanishes, because its domain is U 0 U 1, thus not containing p, which is the only zero of φ 0 = z. (3) When working on P 1, using the standard cover {U 0, U 1 } of example 1, we may set a transition function by g 01 = z m, where z = φ 0 are the coordinates on U 0. This way, we obtain a line bundle that we denote by O(m). When E is a vector bundle over P 1, we use E(m) to denote the vector bundle E O(m). (4) The canonical bundle K of a Riemann surface S is a line bundle given by the holomorphic 1-forms over it. Each holomorphic 1-form is locally given by ω U = f(z)dz, so when we change coordinates to z = φ V φ 1 (z), so does change its local expression: U ω V = f(z) dz d z d z = f( z) (φ U φ 1 V ) ( z) d z. It immediately follows that the transition functions of K can be given by: 1 g V U (z) = (φ U φ 1 V ) ( z) = (φ V φ 1 U ) (z) We can obtain more examples of vector bundles through certain operations. We shall consider: the direct sum, the dual, the tensor product and the determinant line bundle of vector bundles. (5) Consider E and F are vector bundles with rank m and n (respectively) and transition functions A ij and B ij (respectively), for a covering {U i } of S. We define the direct sum of E and F as the vector bundle E F with rank m + n, whose transition function are given by: [ ] Aij 0 C ij :=. 0 B ij 6

17 (6) The dual of E, as set above, is the vector bundle E with rank m, whose transition functions are given by: A ij := (A 1 ij )T. (The superscript T is used for the transpose of a matrix.) (7) The tensor product of E and F, as set above, is the vector bundle E F with rank mn, whose transition functions are given by: C ij (u j v j ) := u i v i = (A ij u j ) (B ij v j ) This defines C ij (x) for a basis of E x F x, so that we may define it everywhere by linearity. (8) The determinant line bundle of E, as set above, is the line bundle det(e), whose transition functions are given by: c ij := det(a ij ) Another motivation for this example is to use the highest exterior power forms on the fibers of E, thus we may may also denote it as m E = det(e). Definition 7. A holomorphic section of a vector bundle E, over a Riemann surface S, is a holomorphic map s : S E satisfying: π s = id S Example 4. (1) Every vector bundle has at least one section. Namely the one whose value at every point of S is 0. (2) The sections of a trivial line bundle over S are in one-to-one correspondence with the holomorphic functions on S. (3) L p always has a non-zero holomorphic section denoted by s p, with only one zero at p. It is defined by: ϕ 0 s p = (id, z), on U 0 ϕ 1 s p = (id, 1), on U 1 One can see that this agrees with the transition function: ϕ 0 s p = (ϕ 0 ϕ 1 1 ) (ϕ 1 s p ) = (ϕ 0 ϕ 1 1 ) (id, 1) = (id, g 01 1) = (id, z) 7

18 (4) The sections of the canonical line bundle K, are precisely the holomorphic 1-forms on S. Remark. The set of all holomorphic sections of a vector bundle E over S is denoted by H 0 (S, E). On the next proposition we determine the set of holomorphic sections of the line bundle O(m), m N. Proposition 2. The set of holomorphic sections of the line bundle O(m), m N, over the Riemann sphere satisfies: dim C H 0 (P 1, O(m)) = { 0, if m < 0 m + 1, if m 0 Proof. For simplicity, we denote by z the local coordinates in U 0 and by z the local coordinates in U 1. So, for a point p P 1 we write z(p) = φ 0 (p) and z(p) = φ 1 (p). Let us consider a general section s H 0 (P 1, O(m)). Write s 0 : C C for the local coordinates of s in U 0. As s 0 is holomorphic it can be expressed by a Laurent series: s 0 (z) = + k=0 a k z k On the other hand, the local coordinates of s in U 1, s 1 : C C, should also describe an holomorphic function. For z C = φ 1 (U 0 U 1 ), we have: s 1 ( z) = g 10 (z)s 0 (z) = 1 g 01 (z) s 0(z) = ( ) k + z m a k z k = z m a k = a k z 1 z m k k=0 k=0 k=0 The above series is the Laurent series of s 1 around z = 0. As it has to be defined at z = 0, it follows, only the terms with a non-negative power are admissible. Thus, a k = 0, k > m. So, if m < 0, the section must vanish everywhere and if m 0, we have m + 1 degrees of freedom for the numbers a 0,..., a m. Below we define a morphism between vector bundles. classifying vector bundles up to isomorphism. It will be useful again when Definition 8. A morphism between the vector bundles π 1 : E 1 S 1 and π 2 : E 2 S 2 is a pair of functions ϕ : E 1 E 2 and f : S 1 S 2, such that the following diagram commutes: ϕ E 1.. E 2 π 1 π f S 1.. S 2 8

19 and ϕ induces a linear map between the fibers π1 1 (x) and π 1 2 (g(x)), x S 1. When ϕ and f describe an invertible morphism we say that they are an isomorphism. 1.4 Presheaves and Sheaves Another concept that is very useful when talking about vector bundles is sheaves. In this context, we often use a particular sheaf, the sheaf of sections, which is obtained from associating each open set U S to the sections of the vector bundle that are holomorphic in U. As usual, we start by defining presheaf. Definition 9. Let S be a Riemann surface. A presheaf of abelian groups F over S, is an assignment of an abelian group F(U) to each open set U S, together with morphisms ρ V U : F(V ) F(U), whenever U V, such that: i) ρ UU = id F(U) ; ii) If U V W, then ρ V U ρ W V = ρ W U. Remark. For an open set U S, we generally designate the elements of F(U) as the sections of F on U. We may also use Γ(U, F) to denote the group F(U). When U V S are open sets and s F(V ) is a section over V, we may use s U to denote ρ V U (s). Having defined a presheaf, we impose some restrictions to it, which will give us a sheaf. Definition 10. A presheaf F over S is a sheaf if, in addition to the conditions of a presheaf, it satisfies: iii) If s F(V ) and {U i } is a covering of V, such that ρ V Ui (s) = 0, i then s = 0; iv) If {U i } is a covering of V and s i F(U i ) agree on the intersections, that is s i Ui U j = s j Ui U j, i, j then it exists s F(V ), such that s Ui = s i. Example 5. Consider E a vector bundle over S. We obtain the sheaf of sections of E, which is denoted by O(E), by associating to each open set U S the set of holomorphic sections on U. For the morphisms we simply use the restriction of the sections. Example 6. In a similar way, we define the sheaf of holomorphic functions, O S, associating to each open set of S the holomorphic functions on it. If G is a set with a complex structure (or a complex manifold), we may also define a sheaf of the functions from open sets of S to G that are locally holomorphic, which is denoted by O S (G). Again, for each of this cases, the morphisms of the sheaves are the restriction of functions. 9

20 An important concept used to describe sheaves (or presheaves) are the stalks F p, p S, of a sheaf (or presheaf) F. These are defined through the sets of sections F(U) of successively smaller neighborhoods U of p S: F p = / F(U) p U where for s 0 F(U 0 ) and s 1 F(U 1 ), s 0 s 1, if and only if it exists V U 0 U 1, such that s 0 V = s 1 V. Other important fact is that for any presheaf F, there exists a sheaf associated to it, F +, which is unique up to isomorphism. In particular, one may see that the group of stalks for both a preasheaf F and the sheaf associated to F are the same. For a more detailed description of this construction we recommend the reader to check [3]. Example 7. One of the simplest examples of sheaf is called the skyscraper sheaf. Given a group G and point p S, a skyscraper sheaf, G, is one whose sections are given by: { 0, if p U G(U) = G, if p U Its easy to check that for every point q S different from p the stalk of G at q is G q = {0}, and the stalk at p is G p = G. 1.5 Cohomology Groups To construct the cohomology groups of a given sheaf F over S we will use a general covering U = {U α }. A p-cochain is an assignment of every intersection of p + 1 open sets U α0,..., U αp to an element g α0...α p F(U α0 U αp ). When the order of the sets is changed, we also change g α0...α p = sgn(σ) g σ(α0...α p) according to the sign of the permutation. The set of all p-cochains for the covering U is denoted by C p (U, F). We define the coboundary map : C p (U, F) C p+1 (U, F) for g C p (U, F). p+1 ( g) α0...α p+1 = ( 1) k g α0... α k...α p+1 k=0 Inside the set of p-cochains we consider two subsets: the set of p-cocycles, Z p (U, F), and the set of p-coboundaries, B p (U, F). A p-cocycle is a p-cochain g, such that g = 0. A p- coboundary g which is in the image of, that is it exists f C p 1 (U, F) satisfying f = g. One can easily see that every B p (U, F) Z p (U, F), as for every g = f B p 1 (U, F), we 10

22 1.6 Exact Sequences In a similar way that we have exact sequences of groups, we may also have exact sequences of sheaves. In short words, we say that a sequence of sheaves over S, 0 R S T 0 is exact if for any p S there is a neighborhood V of p, such that whenever p U V, 0 R(U) S(U) T (U) 0 is a short exact sequence. Moreover, any short exact sequence of sheaves defines a long exact sequence with the cohomology groups, 0 H 0 (S, R) H 0 (S, S) H 0 (S, T ) H 1 (S, R) The concept of exact sequence of sheaves comes in handy in many exercises and proofs, and so it is important to have a good intuition for it. Consider the following exact sequence of sheaves: We obtain part of a long exact sequence: 0 O S (Z) O S exp(2πif) O S = O S (C ) 1 (1.1) H 1 (S, O S) δ H 2 (S, O S (Z)) Using Poincaré Duality we get H 2 (S, O S (Z)) = H 0 (S, Z) = Z: Definition 11. The degree of a line bundle L, deg(l) is defined as the integer δ([l]) (it may also be denoted by c 1 (L), the first Chern class). For a vector bundle E, we define its degree as deg(det(e)). Remark. We will see later that when S = P 1, δ is an isomorphism. So, in the Riemann sphere, a line bundle is uniquely defined by its degree. Corollary 1. In the Riemann sphere L p = O(1) and, as a consequence, L m p = O(m). Proof. It is enough to check that both L p and O(1) are line bundles with degree 1. 12

23 1.7 Fundamental Results Before proceeding, we wish to introduce the following results. They are: Riemann-Roch s Theorem and Serre s Duality. These are fundamental results in Algebraic Geometry and we will use them a lot in the sequel. Definition 12. The genus of a Riemann surface is the integer dim C H 0 (S, K) (where K is the canonical bundle). In general, it is denoted by g. Remark. When E is a vector bundle over S, for simplification, we use h p (S, E) to denote the positive integer dim C H p (S, E), with p N. Theorem 1 (Riemann-Roch). Let S be a compact connected Riemann surface with genus g and E a vector bundle over S. We have: h 0 (S, E) h 1 (S, E) = deg E + (1 g) rk E Proof. We recommend the reader to check the proof on [4]. Theorem 2 (Serre s Duality). Let E be a vector bundle over S. We have: H 1 (S, E) = H 0 (S, E K) Proof. For a proof, see, for example, [6]. Corollary 2 (Riemann-Roch, alternative form). Let S be a compact connected Riemann surface with genus g and E a vector bundle over S. We have: h 0 (S, E) h 0 (S, E K) = deg E + (1 g) rk E Proof. It is a simple application of Serre s Duality to Riemann-Roch s Theorem. The term h 1 (S, E) in the latter is exchanged by h 0 (S, E K) because, using Serre s Duality, we have: dim C H 1 (S, E) = dim C H 0 (S, E K) = dim C H 0 (S, E K) We are now in condition of proving a result we mentioned before. Proposition 4. The map δ that gives the degree of a line bundle over P 1 is a bijection. Proof. From the equation 1.1 we obtain the following sequence of cohomology groups: H 1 (P 1, O P 1) H 1 (P 1, OP 1) δ H 2 (P 1, O P 1(Z)) = Z H 2 (P 1, O P 1) and we will see that the extremities are zero. It is well known that H p (P 1, O P 1) = 0, p 2. This settles the term on the right. 13

24 As for the left term, we begin by using Serre s Duality to obtain dim C H 1 (P 1, O P 1) = dim C H 0 (P 1, K). By definition, the last expression is the genus, g, which corresponds to the number of holes of the Riemann sphere, and so is equal to zero. However, we will also determine the genus according to the above definition. First, using the computations on example 3.4, we see that K corresponds to the line bundle obtained from the standard two chart covering U 0, U 1, and with transition function g 01 (z) = z 2. Hence, K = O( 2) and by Proposition 2, g = 0. Therefore, δ must be an isomorphism. 14

29 Chapter 2 Principal Bundles On this chapter we present Grothendieck s Theorem for principal bundles. We start by defining fiber bundle and principal bundle. Then, we give a description of isomorphism classes of principal bundles using cohomology. For the latter, we will first introduce a description of the first cohomology group for sheaves of non-abelian groups. After this, we finally present Grothendieck s Theorem and show the vector bundle case as a corollary of the previous. We end the chapter with a proof of Grothendieck s Theorem for vector bundles that have a holomorphic quadratic from, or symplectic form defined on its fibers. Definition 13. A fiber bundle over a Riemann surface S is a topological space E, together with a surjective map π : E S, called projection, and a topological space F, called fiber, such that for every p S there is a neighborhood U S, such that the following diagram commutes: π 1 φ (U).. U F.. U π.. (φ is a homeomorphism and p 1 is the usual projection on the first coordinate). Definition 14. A principal bundle E with structure group G (or simply G-bundle) is a fiber bundle with a variety E and fiber G (G is a topological group), together with a continuous right action of G on E. The action of G preserves each fiber of E and its restriction to π 1 (p), p S, is free and transitive. Example 8. (1) The trivial bundle S F is the simplest example of a fiber bundle over S, with fiber F. 19 p 1

30 (2) Vector bundles can also be given as principal bundles with structure group GL(n, C) (see Definition 16). Definition 15. A morphism between two G-bundles π 1 : E 1 S 1 and π 2 : E 2 S 2 is a pair of functions ϕ : E 1 E 2 and f : S 1 S 2, such that the following diagram commutes: ϕ E 1.. E 2 π 1 π f S 1.. S 2 and the function ϕ induces a G-equivariant map between the fibers π1 1 (p) and π 1 2 (f(p)), p S Cohomology of non-abelian Groups We wish to give another description of a vector bundle which involves cohomology. Remember that the transition functions of a vector bundles take values in GL(n, C), which is not an Abelian group. That is why we want a more general description of cohomology group (in fact, this will only apply directly for the first cohomology group). Let us start by considering a sheaf of non-abelian groups F over S and a covering U = {U i }. As mentioned before, a 1-cochain is an assignment of every intersection of 2 open sets U i, U j to an element g ij F(U i U j ). When the order of the sets is changed we multiply g ij = ( 1)g ji by 1. A 1-cocycle is a 1-cochain g, such that: g ij g jk = g ik g jk g 1 ik g ij = 1, in U ijk, i, j, k The set of 1-cocycles is denoted by Z 1 (U, F) and, in general, it is not a subgroup of the set of 1-cochains, C 1 (U, F). We define an action of C 0 (U, F) in C 1 (U, F) by: (D(g 0 ) g 1 ) ij = g i g ij g 1 j, i, j, where g 0 = (g i ) C 0 (U, F) g 1 = (g ij ) C 1 (U, F) Doing a simple calculation, one concludes that Z 1 (U, F) is stable under the action of D(C 0 (U, F)). So, we are able to define the first cohomology group of F with respect to U, as the set of orbits of Z 1 (U, F) under the action of D(C 0 (U, F)): H 1 (U, F) := Z 1 (U, F)/D(C 0 (U, F)) 20

31 Again, we can obtain the cohomology groups independently of the coverings using successively finer coverings and taking the direct limit: H 1 (S, F) := lim U H 1 (U, F) Now, when we consider a vector bundle E, we can think of its transition functions (A ij ) C 1 (U, O S (GL(n, C))) as 1-cochains, with respect to a covering U = {U i }, which are indeed 1-cocycles because, by definition, we have: A ik = A ij A jk, i, j, k Moreover, if we change the local trivializations multiplying them by (A i ) C 0 (U, O S (GL(n, C))), we are simply changing our local referential for the fibers, so we expect to obtain an homeomorphic vector bundle Ẽ with transition functions given by: Ã ij = A i A ij A 1 j Thus, if we identify Ẽ E it is natural to relate vector bundles with the corresponding class of H 1 (S, O S (GL(n, C))). Definition 16 (alternative). A vector bundle of rank n over a Riemann surface S is a class in H 1 (S, O S (GL(n, C))). In particular a line bundle can also be given as a class in H 1 (S, O S (C )). A completely analogous approach is also valid for G-bundles in general, using its transition functions (g ij ) C 1 (U, O S (G)), obtained from: ϕ i ϕ 1 j (Ui U j) G : (U i U j ) G (U i U j ) G (p, g) (p, g ij (p) g) Definition 17 (alternative). A principal bundle with structure group G over a Riemann surface S is a class in H 1 (S, O S (G))). 21

32 2.2 Complex Reductive Lie Groups We are now preparing to look at Grothendieck s general classification theorem. Grothendieck s Theorem for Vector Bundles will be a consequence of this result, as any complex vector bundle of rank n is a GL(n, C)-bundle. This result provides a different description of the G-bundles over P 1, given a complex reductive Lie group G. A reductive Lie group is one whose Lie algebra is the direct sum of its center with a semi-simple Lie algebra. For the theory of reductive Lie groups we refer to [5]. However, in this work we concentrate on elementary examples of reductive groups such as GL(n, C), O(n, C) and Sp(n, C). Before continuing, we shall fix some notation. We will denote by G the Lie algebra of G and h one of its Cartan algebras. H will denote the Cartan group associated to h. N will denote the normalizer of H inside G and W will denote the Weyl group of G, which is the discrete group given by W = N/H. 2.3 Grothendieck s Theorem for Principal Bundles We will compare principal bundles with structure group G with the ones with structure group H over P 1. To do this we use their description as classes from the cohomology groups H 1 (P 1, O P 1(G)) and H 1 (P 1, O P 1(H)). We can see that conjugation by an element of N stabilizes H G, so it defines a map on the classes of H 1 (P 1, O P 1(H)). However, from the construction of the cohomology groups we see that conjugation by an element of H acts trivially on the classes of H 1 (P 1, O P 1(H)), so we can say that the Weyl group W = N/H acts on H 1 (P 1, O P 1(H)) through conjugation. The inclusion H G also induces a map H 1 (P 1, O P 1(H)) H 1 (P 1, O P 1(G)). Furthermore, as N G, conjugation with an element of N will act trivially on the classes of H 1 (P 1, O P 1(G)) and so the following map is well defined: H 1 (P 1, O P 1(H))/W H 1 (P 1, O P 1(G)) (2.1) The main result of the article is the following: Theorem 4 (Grothendieck). The map defined in (2.1), induced by the inclusion of groups, is bijective. Here, we will not prove this theorem, and refer to [2]. 2.4 Application to Vector Bundles The classification of vector bundles over the Riemann sphere, as given in Theorem 3, can be also obtained as a corollary of the previous, more general result. 22

33 Corollary 3. For any vector bundle E of rank n over P 1, there are integers a 1,..., a n, unique up to permutation, such that E decomposes as E = O(a 1 ) O(a n ) Proof. Remember that vector bundles of rank n are the principal bundles with structure group GL(n, C). So, we take G = GL(n, C) on the Gothendieck s general Theorem. The Cartan Subgroup of this group is given by the set of invertible diagonal matrices, thus we get H = (C ) n. Now, we apply the same Theorem: H 1 (P 1, O P 1(GL(n, C))) = H 1 (P 1, O P 1(H))/W = (H 1 (P 1, O P 1(C )) n )/W and the isomorphism is induced by the inclusion H G. Now, we will show that the Weyl group acts on H 1 (P 1, O P 1(C )) n through permutation of the line bundles that we use on the direct sum. As the factors on the statement of Grothendieck s Theorem for Vector Bundles are unique up to permutation, this will conclude our proof. By definition W = N/H, where N is the normalizer of the Cartan subgroup H. g N if and only g 1 Dg = D for every diagonal matrix D. Consider the case where the only non-zero entry of D is the entry (i, i), assuming the value 1. We can see that all entries of Dg, for the exception of the i-th line, are zero. As g is invertible and g 1 (Dg) = D (remember that D has only one non-zero entry), we conclude that exactly one coordinate of the vector (Dg) i = g i is non-zero. Therefore, every matrix in N has the form P D, where P is a permutation matrix and D is an invertible diagonal matrix. It is now obvious that we may choose permutation matrices as representatives of the classes on W = N/H and so W = S n, ending the proof. 23

35 2.6 When G = O(n, C) From the inclusion O(n, C) GL(n, C) we may see that every principal bundle with structure group O(n, C) also defines a vector bundle. Furthermore, a vector bundle has a nondegenerate holomorphic symmetric bilinear form (quadratic form) on its fibers, if and only if it can be reduced to a O(n, C)-bundle. On this section we will provide a classification result for O(n, C)-bundles. The following result shows that two distinct O(n, C)-bundles define distinct vector bundles, or equivalently, a quadratic form on the fibers of a vector bundle is unique up to a linear transformation on the fibers. Proposition 5. The map H 1 (P 1, O P 1(O(n, C))) H 1 (P 1, O P 1(GL(n, C))) is injective. Proof. Let us consider a fiber bundle E H 1 (P 1, O P 1(GL(n, C))) which is the image of two fiber bundles A, B H 1 (P 1, O P 1(O(n, C))) through the inclusion map. Let us also denote by A ij and B ij the transition functions of A and B, respectively. As the fiber bundles A and B coincide as vector bundles, there are matrices C i Γ(U i, O P 1(GL(n, C))) such that: C i A ij C 1 j = B ij C i = B ij C j A 1 ij (2.2) Our objective is to prove that it also exists matrices C i Γ(U i, O P 1(O(n, C))) such that: C ia ij (C j) 1 = B ij (2.3) so that A and B will define the same class in H 1 (P 1, O P 1(O(n, C))). Using the previous equation, we get: Ci T C i = (B ij C j A 1 ij )T B ij C j A 1 ij = A ij C T j B T ijb ij C j A 1 ij = A ij C T j C j A 1 ij We conclude that the C T C induces a map on the fibers of E, because it agrees with the transition functions. The characteristic polynomial of C T C is also well defined as: det(c T i C i z I) = det(a 1 ij (CT i C i z I)A ij ) = det(c T j C j z I) Furthermore, as its coefficients define an holomorphic function on P 1 and this surface is compact, it follows that the characteristic polynomial is constant. We denote by p(z) this polynomial and by h(z) a polynomial such that h(z) 2 z is divisible by p(z), according to Lemma 2. 25

36 Now, set h(c T i C i) = Q i. By the Cayley-Hamilton Theorem, we have p(c T i C i) = 0, so we may conclude that Q 2 i = h(ct i C i) 2 = C T i C i. As C T i C i is symmetric, its powers will also be symmetric, and so we may conclude that Q i = h(c T i C i) = (h(c T i C i)) T = Q i, i.e. Q i is symmetric. Finally, we verify that C i = C iq 1 i Γ(P 1, O P 1(O(n, C))): (C i) T C i = (Q 1 i ) T Ci T C i Q 1 i = Q 1 i Q 2 i Q 1 i = I n and using the fact that u induces a map on the fibers of E, together with equation (2.2), we have: This ends the proof. C ia ij (C j) 1 = (C i Q 1 i )A ij (Q j C 1 j ) = C i Q 1 i (A ij Q j A 1 ij )A ijc 1 j = C i Q 1 i Q i A ij C 1 j = B ij 2.7 Orthogonal Structure We now present a necessary and sufficient condition as to when it is possible to reduce a vector bundle to a O(n, C)-bundle. Theorem 5. Let E be a vector bundle. E has an orthogonal structure if and only if it is isomorphic to its dual, E. Proof. In the case that E has an orthogonal structure we obtain every homomorphism on the fibers of E from the product by a section of E, using the quadratic form defined on the fibers of E, so it easily follows that E is isomorphic to its dual. In the case that E is isomorphic to E, we can use the factorization of E as line bundles to get: E = O(m 1 ) O(m n ) E = O( m1 ) O( m n ) As this factorization is unique up to isomorphism we conclude that we can pair any term m i with m i. Thus, by setting R = m O(m i>0 i) we have: E = E 0 R R Lemma 3. R R is endowed with a orthogonal structure. 26

38 2.8 When G = Sp(2n, C) As the previous case, from the inclusion Sp(2n, C) GL(2n, C) we may see that every principal bundle with structure group Sp(2n, C) also defines a vector bundle. Furthermore, a vector bundle has a nondegenerate holomorphic antisymmetric bilinear form (symplectic form) on its fibers, if and only if it can be reduced to a Sp(2n, C)-bundle. On this section we will provide a classification result for Sp(2n, C)-bundles. The first result shows that two distinct Sp(2n, C)-bundles define distinct vector bundles, or equivalently, a symplectic form on the fibers of a vector bundle is unique up to a linear transformation on the fibers. Definition 18. A matrix M GL(2n, C) is symplectic if M T ΩM = Ω, Ω = [ ] 0 In I n 0 The set of all (2n) (2n) complex symplectic matrices is denoted by Sp(2n, C). Remark. The matrix Ω satisfies the following relations: 1) Ω T = Ω; 2) Ω 1 = Ω = Ω T. Proposition 6. The map H 1 (P 1, O P 1(Sp(2n, C))) H 1 (P 1, O P 1(GL(2n, C))) is injective. Proof. We shall follow the proof of the previous proposition. Let us consider a vector bundle E H 1 (P 1, O P 1(GL(2n, C))) which is the image of two principal bundles A, B H 1 (P 1, O P 1(Sp(2n, C))), with transition functions A ij and B ij, respectively, for a given cover U = {U i }. As A and B coincide as vector bundles, there are matrices C i Γ(P 1, O P 1(GL(2n, C))), such that: C i A ij C 1 j With some more calculations, we get: = B ij C i = B ij C j A 1 ij ΩC T i ΩC i = Ω(B ij C j A 1 ij )T Ω(B ij C j A 1 ij ) = ( Ω(A 1 ij )T Ω)( Ω)Cj T Ω( ΩBijΩB T ij )C j A 1 ij = A ij ( ΩC T j ΩC j )A 1 ij This way, we may conclude that multiplication by ΩC T ΩC is well defined on the fibers of E (or A), as it agrees with the transition functions. In the same way, the characteristic polynomial of ΩC T ΩC is well defined and it is constant, as its coefficients define global 28

39 holomorphic functions on P 1. We denote by p(z) this polynomial and by h(z) a polynomial such that h(z) 2 z is divisible by p(z), according to Lemma 2. Now, Set Q i = h( ΩC T i ΩC i). By the Cayley-Hamilton Theorem, we have p( ΩC T i ΩC i) = 0, so we may conclude that Q 2 i = ΩCT i ΩC i. As ΩC T i ΩC i satisfies Ω( ΩC T i ΩC i) T Ω = ( ΩC T i ΩC i), its powers will also satisfy that equation. By linearity, it follows that ΩQ T i Ω = Q i. and Furthermore, we can see that C i = C iq 1 i (C i Q 1 i ) T ΩC i Q 1 i Γ(P 1, O P 1(Sp(2n, C))) as: = (Q 1 i ) T Ω( ΩCi T ΩC i )Q 1 i = (Q 1 i ) T ΩQ 2 i Q 1 i = (Q T i ) 1 (Q T i Ω) = Ω C ia ij (C j) 1 = (C i Q 1 i )A ij (Q j C 1 j ) = C i Q 1 i A ij Q j A 1 ij A ijc 1 j = C i Q 1 i Q i A ij C 1 j = C i A ij C 1 j = B ij Finally, we conclude that A and B are the same principal bundle with group structure Sp(2n, C). 2.9 Symplectic Structure We now present a necessary and sufficient condition as to when it is possible to reduce a vector bundle to a Sp(2n, C)-bundle. Theorem 6. Let E be a vector bundle of rank 2n. E has a symplectic structure if and only if it is isomorphic to its dual, E. Proof. As in the case for the orthogonal structure, in the case that E has a symplectic structure we obtain every homomorphism on the fibers of E from the product by a section of E, using the symplectic form defined on the fibers of E, so it easily follows that E is isomorphic to its dual. In the case that E and E are isomorphic, we can use the factorization of E as line bundles to get: E = O(m 1 ) O(m 2n ) E = O( m1 ) O( m 2n ) 29

40 As this factorization is unique up to isomorphism we conclude that we can pair any term m i with m i. Thus, by setting R = m O(m i>0 i), we have: E = E 0 R R Lemma 4. R R is endowed with a symplectic structure. Proof. We define a symplectic form on the fibers of R R given by: (a + a, b + b ) = a, b b, a, a, b R x, a, b Rx, x P 1 Notice that E 0 can also be endowed with a symplectic structure which can be defined by (u, v) = u T Ωv, u, v E x, x P 1, when considering the chart where E 0 = P 1 C 2r. Finally, adding the the symplectic forms from both components of the direct sum gives us a symplectic on the global space E. 30

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